Numerical methods for stochastic Volterra integral equations with weakly singular kernels.

In this paper we first establish the existence, uniqueness and Hölder continuity of the solution to stochastic Volterra integral equations (SVIEs) with weakly singular kernels, with singularities |$\alpha \in (0, 1)$| for the drift term and |$\beta \in (0, 1/2)$| for the stochastic term. Subsequently, we propose a |$\theta $| -Euler–Maruyama scheme and a Milstein scheme to solve the equations numerically and obtain strong rates of convergence for both schemes in |$L^{p}$| norm for any |$p\geqslant 1$|⁠. For the |$\theta $| -Euler–Maruyama scheme the rate is |$\min \big\{1-\alpha ,\frac{1}{2}-\beta \big\}~ $| and for the Milstein scheme is |$\min \{1-\alpha ,1-2\beta \}$|⁠. These results on the rates of convergence are significantly different from those it is similar schemes for the SVIEs with regular kernels. The source of the difficulty is the lack of Itô formula for the equations. To get around this difficulty we use the Taylor formula subsequently carrying out a sophisticated analysis of the equation. [ABSTRACT FROM AUTHOR]